Optimal. Leaf size=227 \[ -\frac{3 (A (3 m+5)+C (3 m+2)) \sin (c+d x) (b \sec (c+d x))^{2/3} \sec ^{m-1}(c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{6} (1-3 m),\frac{1}{6} (7-3 m),\cos ^2(c+d x)\right )}{d (1-3 m) (3 m+5) \sqrt{\sin ^2(c+d x)}}+\frac{3 B \sin (c+d x) (b \sec (c+d x))^{2/3} \sec ^m(c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{6} (-3 m-2),\frac{1}{6} (4-3 m),\cos ^2(c+d x)\right )}{d (3 m+2) \sqrt{\sin ^2(c+d x)}}+\frac{3 C \sin (c+d x) (b \sec (c+d x))^{2/3} \sec ^{m+1}(c+d x)}{d (3 m+5)} \]
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Rubi [A] time = 0.190569, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.122, Rules used = {20, 4047, 3772, 2643, 4046} \[ -\frac{3 (A (3 m+5)+C (3 m+2)) \sin (c+d x) (b \sec (c+d x))^{2/3} \sec ^{m-1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (1-3 m);\frac{1}{6} (7-3 m);\cos ^2(c+d x)\right )}{d (1-3 m) (3 m+5) \sqrt{\sin ^2(c+d x)}}+\frac{3 B \sin (c+d x) (b \sec (c+d x))^{2/3} \sec ^m(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (-3 m-2);\frac{1}{6} (4-3 m);\cos ^2(c+d x)\right )}{d (3 m+2) \sqrt{\sin ^2(c+d x)}}+\frac{3 C \sin (c+d x) (b \sec (c+d x))^{2/3} \sec ^{m+1}(c+d x)}{d (3 m+5)} \]
Antiderivative was successfully verified.
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Rule 20
Rule 4047
Rule 3772
Rule 2643
Rule 4046
Rubi steps
\begin{align*} \int \sec ^m(c+d x) (b \sec (c+d x))^{2/3} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{(b \sec (c+d x))^{2/3} \int \sec ^{\frac{2}{3}+m}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx}{\sec ^{\frac{2}{3}}(c+d x)}\\ &=\frac{(b \sec (c+d x))^{2/3} \int \sec ^{\frac{2}{3}+m}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx}{\sec ^{\frac{2}{3}}(c+d x)}+\frac{\left (B (b \sec (c+d x))^{2/3}\right ) \int \sec ^{\frac{5}{3}+m}(c+d x) \, dx}{\sec ^{\frac{2}{3}}(c+d x)}\\ &=\frac{3 C \sec ^{1+m}(c+d x) (b \sec (c+d x))^{2/3} \sin (c+d x)}{d (5+3 m)}+\frac{\left (\left (C \left (\frac{2}{3}+m\right )+A \left (\frac{5}{3}+m\right )\right ) (b \sec (c+d x))^{2/3}\right ) \int \sec ^{\frac{2}{3}+m}(c+d x) \, dx}{\left (\frac{5}{3}+m\right ) \sec ^{\frac{2}{3}}(c+d x)}+\left (B \cos ^{\frac{2}{3}+m}(c+d x) \sec ^m(c+d x) (b \sec (c+d x))^{2/3}\right ) \int \cos ^{-\frac{5}{3}-m}(c+d x) \, dx\\ &=\frac{3 C \sec ^{1+m}(c+d x) (b \sec (c+d x))^{2/3} \sin (c+d x)}{d (5+3 m)}+\frac{3 B \, _2F_1\left (\frac{1}{2},\frac{1}{6} (-2-3 m);\frac{1}{6} (4-3 m);\cos ^2(c+d x)\right ) \sec ^m(c+d x) (b \sec (c+d x))^{2/3} \sin (c+d x)}{d (2+3 m) \sqrt{\sin ^2(c+d x)}}+\frac{\left (\left (C \left (\frac{2}{3}+m\right )+A \left (\frac{5}{3}+m\right )\right ) \cos ^{\frac{2}{3}+m}(c+d x) \sec ^m(c+d x) (b \sec (c+d x))^{2/3}\right ) \int \cos ^{-\frac{2}{3}-m}(c+d x) \, dx}{\frac{5}{3}+m}\\ &=\frac{3 C \sec ^{1+m}(c+d x) (b \sec (c+d x))^{2/3} \sin (c+d x)}{d (5+3 m)}-\frac{3 (C (2+3 m)+A (5+3 m)) \, _2F_1\left (\frac{1}{2},\frac{1}{6} (1-3 m);\frac{1}{6} (7-3 m);\cos ^2(c+d x)\right ) \sec ^{-1+m}(c+d x) (b \sec (c+d x))^{2/3} \sin (c+d x)}{d (1-3 m) (5+3 m) \sqrt{\sin ^2(c+d x)}}+\frac{3 B \, _2F_1\left (\frac{1}{2},\frac{1}{6} (-2-3 m);\frac{1}{6} (4-3 m);\cos ^2(c+d x)\right ) \sec ^m(c+d x) (b \sec (c+d x))^{2/3} \sin (c+d x)}{d (2+3 m) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [C] time = 6.70575, size = 547, normalized size = 2.41 \[ -\frac{3 i 2^{m+\frac{5}{3}} e^{-\frac{1}{3} i d (3 m+2) x} \left (\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{m+\frac{2}{3}} \left (1+e^{2 i (c+d x)}\right )^{m+\frac{2}{3}} (b \sec (c+d x))^{2/3} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac{A e^{4 i c+\frac{1}{3} i d (3 m+14) x} \text{Hypergeometric2F1}\left (\frac{m}{2}+\frac{7}{3},m+\frac{8}{3},\frac{1}{6} (3 m+20),-e^{2 i (c+d x)}\right )}{3 m+14}+\frac{A e^{\frac{1}{3} i d (3 m+2) x} \text{Hypergeometric2F1}\left (m+\frac{8}{3},\frac{1}{6} (3 m+2),\frac{1}{6} (3 m+8),-e^{2 i (c+d x)}\right )}{3 m+2}+\frac{2 A e^{\frac{1}{3} i (6 c+d (3 m+8) x)} \text{Hypergeometric2F1}\left (m+\frac{8}{3},\frac{1}{6} (3 m+8),\frac{m}{2}+\frac{7}{3},-e^{2 i (c+d x)}\right )}{3 m+8}+\frac{2 B e^{\frac{1}{3} i (3 c+d (3 m+5) x)} \text{Hypergeometric2F1}\left (m+\frac{8}{3},\frac{1}{6} (3 m+5),\frac{1}{6} (3 m+11),-e^{2 i (c+d x)}\right )}{3 m+5}+\frac{2 B e^{\frac{1}{3} i (9 c+d (3 m+11) x)} \text{Hypergeometric2F1}\left (m+\frac{8}{3},\frac{1}{6} (3 m+11),\frac{1}{6} (3 m+17),-e^{2 i (c+d x)}\right )}{3 m+11}+\frac{4 C e^{\frac{1}{3} i (6 c+d (3 m+8) x)} \text{Hypergeometric2F1}\left (m+\frac{8}{3},\frac{1}{6} (3 m+8),\frac{m}{2}+\frac{7}{3},-e^{2 i (c+d x)}\right )}{3 m+8}\right )}{d \sec ^{\frac{8}{3}}(c+d x) (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.18, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( dx+c \right ) \right ) ^{m} \left ( b\sec \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}} \left ( A+B\sec \left ( dx+c \right ) +C \left ( \sec \left ( dx+c \right ) \right ) ^{2} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac{2}{3}} \sec \left (d x + c\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac{2}{3}} \sec \left (d x + c\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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